I have these particular exercise that i cannot solve. I know i have to change the variables, but i cannot figure out if i should use polar coords or any other change.
Let D be the region delimited by:
$$
D = \{(x,y) \in \mathbb{R} ^{2} : (x-1)y \geq 0, \frac{(x-1)^2}{9} + \frac{y^2}{25} \leq 1 \}
$$
Calculate:
$$
\iint\limits_D \sin((x-1)^2 + \frac{9y^2}{25}) \,dxdy
$$
I've tried using $u = \frac{(x-1)}{3}$ and $v = \frac{y}{5}$ so that i can replace in the integral the following:
$$
\iint\limits_D \sin(9(u^2 + v^2)) \frac{1}{15} \,dudv
$$
knowing the Jacobian is $J(x,y)=\frac{\partial (u,v)}{\partial (x,y)}=\frac{1}{15}$.
But i don't know where to follow, or if the variable changes i've made are correct. Can I use that $u^2 + v^2 = 1$, or that's just for polar coords?
Thanks a lot for your help!
Best Answer
You can indeed say now the the region (in $uv$-coordinates) is $D=\{(u,v)\in\mathbb{R}^2|u^2+v^2\leq 1\}$. So you've transformed your original elliptical region into a circular region using your $uv$-change of variable. Treating this as a problem in its own right, you are completely free to change now to polar coordinates.
If you think about it this makes perfect sense. Your first change of variable from $xy$ to $uv$ was a linear map. You can think about a linear map as stretching/compressing the $x$-axis and $y$-axis by some scalar factors and also changing the angle between them. Your transformation does this so that the elliptical region looks circular.